A statistically important Gaussian Process

Abstract

It is well known, that under the condition LAN and some more regularity conditions, the process of log-likelihood functions converges weakly to a degenerate Gaussian process (the trajectories of which are straight lines). In the non-regular case considered by several authors [1, 9] the limiting process is non-degenerate and characterized by the covariance function R(t,s) = 1 2 (|t| p + |s| p − |t − s| p) . In the present paper, we derive sever properties of this process with relevance to applications in statistics. In particular, a bound for the risk of equivariant estimates is given and the maximum likelihood estimate (MLE) is shown to be well defined. The calculation of the exact distribution of the MLE is left as an open problem.